Manhattan GMAT Challenge Problem of the Week – 13 August 2012

by on August 13th, 2012

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Question

Jean puts N identical cubes, the sides of which are 1 inch long, inside a rectangular box, each side of which is longer than 1 inch, such that the box is completely filled with no gaps and no cubes left over. What is N?

(1) 56 < N < 63
(2) N is a multiple of 3.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Answer

To be able to put N cubes into a rectangular box with no gaps and no left-over cubes, you must have the following equality: N = length × width × height. Moreover, if the length, width, and height are all greater than 1, then it must be true that N is the product of at least 3 primes. If N is itself prime or the product of just 2 primes (unique or not), then the condition fails.

So you can rephrase the question this way: is N the product of at least 3 primes?

Statement 1: SUFFICIENT. Since N must be an integer (it counts a set of objects), you can rephrase the statement: N is either 57, 58, 59, 60, 61, or 62. Now test each of these numbers.
57 = 3 × 19 = product of just 2 primes, so 57 is out.
58 = 2 × 29 = product of just 2 primes, so 58 is out.
59 is itself prime, so 59 is out.
60 = 2^2 × 3 × 5 = product of 4 primes, so 60 can work. For instance, the dimensions of the box could be 4 by 3 by 5, or various other combinations of the primes.
61 is prime, so 61 is out.
62 = 2 × 31 = product of just 2 primes, so 62 is out.
Thus, the statement tells us that N must be 60.

Statement 2: NOT SUFFICIENT. It’s easy to come up with multiple examples that fit the conditions. For instance, we can reuse N = 60 (just be careful whenever you reuse a case from one statement in another statement), but then try N = 27 (= 3 × 3 × 3), which also fits. Many multiples of 3 can be legal values of N, so this statement is not enough.

The correct answer is A.

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8 comments

  • Why does N have to be the product of 3 primes?

    • Dear Shane,

      Please let me know once u get ans to ur query..as I too have same query.

  • N need not be product of 3 primes. But it has to be product of atleast 3 nos(length, width,height). So it has to be atleast product of 3  primes.

  • why it is necessary that- lenght - width and height have to be prime numbers.

  • Shiva-

    I think if lenght, width and height are any numbers then there will be multiple possibilities of data set and even "A" will not be correct choice

  • The answer has to be E -- there is no constraint set for l w h other than each is > 1. If the question had specified that l, w, h must by integers then sure, the answer is A.

    • I started with ather simple approach with the minimum number of cubes needed to arrange in a rectangular fashion with dimesnsions more than 1 inch on all sides. The minimum number of cubes required for such an arrangement is 8 and stacks can be added upon it(ofcourse the total number of cubes is always a multiple of 4). Given condition I, the only number fitting this situation is 60.

      Ofcourse on the contrary the condition II doesn't fit at all.

  • Statement 1: The product has to be product of 3 numbers and none of those 3 numbers can be 1 because each of the lengths are > 1, therefore.
    57 - 3*19*1 OUT
    58 - 2*29*1 OUT
    59 - 59*1*1 OUT
    60 - 3*4*5 all sides greater than 1 and so this is a possibility
    61 - 61*1*1 OUT
    62 - 2*31*1 OUT..

    Statement 2 : Too Open ended NSF!! 
    A alone is the answer. 

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